A Variational Computational-based Framework for Unsteady Incompressible Flows

TL;DR

This work develops a variational framework for unsteady incompressible flows by applying the Principle of Minimum Pressure Gradient (PMPG) and solving at each time step with Physics-Informed Neural Networks (PINNs). By minimizing the $L^2$ norm of the pressure gradient, i.e., $\mathcal{A} = \frac{1}{2} \rho \int_{\Omega} \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} - \nu \nabla^2 \mathbf{u}\right)^2 d\mathbf{x}$, under $\nabla\cdot\mathbf{u}=0$, the NSE are recovered as the first-order condition, enabling a pressure-free, time-stepping evolution. The PMPG-PINN is shown to converge faster and handle outflow boundary issues more robustly than conventional PINNs, demonstrated across Poiseuille flow, lid-driven cavity, and flow over a cylinder. The method eliminates Poisson-pressure solves, reduces reliance on artificial outlet conditions, and supports accurate transient predictions with transfer-learning–assisted efficiency gains. Overall, PMPG-PINN offers a computationally efficient alternative for complex unsteady fluid problems with potential for large-scale applications.

Abstract

Advancements in computational fluid mechanics have largely relied on Newtonian frameworks, particularly through the direct simulation of Navier-Stokes equations. In this work, we propose an alternative computational framework that employs variational methods, specifically by leveraging the principle of minimum pressure gradient, which turns the fluid mechanics problem into a minimization problem whose solution can be used to predict the flow field in unsteady incompressible viscous flows. This method exhibits two particulary intriguing properties. First, it circumvents the chronic issues of pressure-velocity coupling in incompressible flows, which often dominates the computational cost in computational fluid dynamics (CFD). Second, this method eliminates the reliance on unphysical assumptions at the outflow boundary, addressing another longstanding challenge in CFD. We apply this framework to three benchmark examples across a range of Reynolds numbers: (i) unsteady flow field in a lid-driven cavity, (ii) Poiseuille flow, and (iii) flow past a circular cylinder. The minimization framework is carried out using a physics-informed neural network (PINN), which integrates the underlying physical principles directly into the training of the model. The results from the proposed method are validated against high-fidelity CFD simulations, showing an excellent agreement. Comparison of the proposed variational method to the conventional method, wherein PINNs is directly applied to solve Navier-Stokes Equations, reveals that the proposed method outperforms conventional PINNs in terms of both convergence rate and time, demonstrating its potential for solving complex fluid mechanics problems.

Figures (13)

• Figure 1: Schematic of a physics-informed neural network based on the unsteady PMPG formulation
• Figure 2: Illustration of the numerical integration algorithm towards 2D randomly distributed sample points.
• Figure 3: Schematic of a physics-informed neural network based on unsteady NS formulation
• Figure 4: Schematic diagram of the two-dimensional (a) Poiseuille flow, (b) lid-driven cavity problem, and (c) the flow past a circular cylinder.
• Figure 5: Results from the minimization training. (a) Residual loss function, $\mathcal{L}_c$. (b) Mean squared error of the predicted $\mathbf{u}_t$ and the reference values, (c) color map of the predicted $\mathbf{u_t}$, and (d) comparison of the predicted $\mathbf{u_t}$ (line) with reference value (dashed lines) at the outlet boundary.